JEE Main 2025 Apr 4 Shift 1, Mathematics Q18: Quadratic Equations

JEE Main2025Apr 4, Shift 1Mathematics

Consider the equation $x^{2} + 4 x - n = 0$ , where $n \in [20, 100]$ is a natural number. Then the number of all distinct values of $n$ , for which the given equation has integral roots, is equal to

Solution

Complete the square: $(x + 2)^{2} = n + 4$ . For $x$ to be an integer, $n + 4$ must be a perfect square.

$n \in [20, 100] \Rightarrow n + 4 \in [24, 104]$ . Perfect squares in this range: $25, 36, 49, 64, 81, 100$ corresponding to $n + 4 = 5, 6, 7, 8, 9, 10$ . That is 6 values of $n$ (namely $21, 32, 45, 60, 77, 96$ ).

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JEE Main 2025 Apr 4 Shift 1, Mathematics Q18: Quadratic Equations

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